Cut angles for piezoelectric quartz crystal elements

ABSTRACT

A piezoelectric crystal element comprising a member of monocrystalline material of symmetry class 32 has two force introduction surfaces essentially parallel to a crystallographic `a`-axis but inclined with respect to the `c`-axis so as to intersect the latter at a specific angle e.g. 10° to 40° to considerably reduce over a desired temperature range the temperature dependence of transverse of antiaxial shear sensitivity.

This application is a continuation application of my copendingapplication Ser. No. 234,702 filed Mar. 15, 1972, now abandoned.

It is known that piezoelectric measuring technology has developed intoone of the most accurate and universal methods of analyzing dynamicforce and pressure processes, accelerations and vibration states.Piezoelectric measuring transducers are distinguished from othermeasuring elements in particular by their very high resonant frequency,extreme rigidity and small dimensions. This enables virtuallytranslationless measurements to be made within minimum reaction on theobject being measured, together with the simultaneous resolution of morethan one vector component. There is the unique facility of being able toapply compensation with static preloads exceeding a measured value bymany orders of magnitude when measuring force, shear and pressure,without affecting precision. Further advantages of piezoelectricmeasuring techniques are the outstanding linearity and lack ofhysteresis over measuring ranges which may extend over several orders ofmagnitude.

The active elements of piezoelectric precision measuring cells consistvery often of quartz crystals, because this material is predestined forsuch applications thanks to its excellent mechanical and electricalproperties. Although many ferroelectric substances with much greaterpiezoelectric effects are available as transducer materials, at thepresent time these do not appear to be capable of providing substitutesfor quartz in applications where high precision is demanded, due totheir inadequate long-time stability and inherent hysteresis.

With the development of high-temperature-resistant materials for gasturbine, nuclear and rocket engineering, a number of problems havearisen with respect to piezoelectric measuring technology, requiringmeasuring cells able of operating reliably over temperature intervalsexceeding 400° C. For such uses the effect of the temperature-dependenceof the piezoelectric sensitivity of quartz is particularly troublesome.Moreover, in view of the present trend favoring rationally calibratedmeasuring chains the temperature effect presents a major complication.The piezoelectric charge coefficient d₁₁, on which the sensitivity ofcontemporary longitudinal, transverse and shear transducer elementsdepends, diminishes by 4.54% between 0 and 200° C. for example, and by9.12% between 200 and 400° C. As may be seen from FIG. 5, thesensitivity drops increasingly at still higher temperatures, finallyreaching zero at the high-low transformation point of quartz (573.3°C.). For this reason the use of transducers with the usual quartzelements is mostly confined to a relatively narrow temperature range.

Devices employed to eliminate the influences of temperature on themeasured result in designs known hitherto were concerned, as a rule,only with compensating pseudo-pyroelecticity, in other words,compensating the differential thermal expansion between the crystalassembly and the transducer housing or preloading system as the casemight be. For this, the crystal assembly is shimmed with compensatingdiscs of non-piezoelectric materials having a higher coefficient ofexpansion than the quartz elements. These familiar configurations are,however, inevitably incapable of compensating deviations in thesensitivity of the crystal elements due to the temperature-dependence ofthe piezoelectric coefficient.

Proposals have been made for compensating this temperature-dependence byelectronic means. For this purpose a thermocouple, thermistor or othertemperature-dependent component is incorporated in the transducer andlinked with the charge amplifier, whose gain it controls, via ameasuring amplifier followed by a function converter. Secondarycompensation systems of this kind have not found acceptance in practice,because they entail a number of serious disadvantages, such as theincreased dimensions of the measuring cell owing to thetemperature-dependent component and its connections, the complication ofadditional electrical conductors, the problems associated with chargeamplifiers controlled through a function converter, negative feedbackpreventing full utilization of the precision of the piezoelectrictransducer elements, etc.

The invention forming the subject of this application serves the purposeof providing piezoelectric crystal elements in which thetemperature-dependence of the piezoelectric sensitivity can be adaptedto requirements by selecting a suitable crystallographic orientation. Aswill be demonstrated below, with the piezoelectric material constants itis possible to calculate the orientation in relation to thecrystallographic axes with which a transducer element must be cut if thetemperature-dependence of its transverse or shear sensitivity is tocoincide optimally with ideal behavior, freely selectable within certainlimits.

Consequently, the present invention enables piezoelectric measuringtransducers to be constructed with virtually a constant sensitivity overa very wide temperature range, without incorporating any temperaturesensors in the measuring cell and without complicating the evaluationelectronics. This improvement enhances the effective measuring precisiondecisively, as well as the functional reliability. Moreover, substantialadvantages emerge with regard to manufacture and stocking, because,apart from the transducer crystals, all components of the measuringcells do not vary with the amount of temperture compensation.

To aid in understanding the present invention, fundamental differencesbetween oscillator quartzes and transducer quartzes will be explained.Oscillator quartzes are electromechanical resonators, in other words,components serving to constantly maintain an exactly defined oscillationfrequency, which is a function of the mechanical dimensions, density andelastic material constants. Temperature-compensated oscillating quartzsections have been known for some time. Their orientation is accordinglychosen so that influences of the temperature-dependence of the elasticconstants and of the thermal expansion on the oscillation frequencycancel each other. On the other hand the temperature dependence ofpiezoelectric coefficients is of negligible importance in resonators,since they are of only subordinate consequence as drive constants. Withtransducer crystals, however, the situation is reversed. These devicesserve to convert mostly aperiodic mechanical signals into electricalsignals and are generally operated far below their resonant frequencies.In this range their temperature-dependence is relatively unimportant. Ofcritical importance for the transducer sensitivity, on the other hand,are the piezoelectric coefficients, whose dependence on temperature iscompensated by this invention.

For a more complete understanding of the present invention, attention isdirected to the accompanying drawings (and description thereof) wherein:

FIG. 1 shows the arrangement of Cartesian coordinates X, Y, Z inrelation to crystal axes a₁, a₂, a₃ and c;

FIG. 2 shows the orientation of a piezoelectric crystal element inrelation to the coordinate axes with a rotation α about the X axis;

FIG. 3 illustrates a twofold rotation, first about the X axis and thenabout the Y' axis;

FIG. 4 illustrates a threefold rotation, first about the X axis, thenabout the Y' axis and finally the Z" axis;

FIG. 5 shows the temperature-dependence of the piezoelectriccoefficients d₁₁ and -d₁₄ of quartz;

FIG. 6 shows the temperature-dependence of the transformed piezoelectriccoefficient -d'₁₂ for crystal elements whose cutting directions arerotated about the X-axis by various angles α.

FIG. 7 is a plot of the temperatures at which the transverse sensitivitycoefficient -d'₁₂ reaches a maximum, versus the orientation angle α.Further the temperature limits of a ± 1% interval of thermal sensitivitydeviation are indicated by dashed lines.

FIG. 8 shows the magnitude of the temperature interval within which thepiezoelectric transversal sensitivity of an X-rotated quartz elementremains constant to within ± 1%, as a function of the angle α.

FIG. 9 shows a comparison of the temperature-dependence of thepiezoelectric sensitivity -d'₁₂ for a section in accordance with theinvention, and the sensitivity d₁₁ for a usual longitudinal section; &

FIG. 10 shows the temperature-dependence of the piezoelectriccoefficient -d'₂₆ for the antiaxial shear sensitivity of quartz elementsrotated about the X - axis by various orientation angles α.

The basis on which the invention is founded may be summarized asfollows. The general equation of state for the direct piezoelectriceffect is: ##EQU1## in which, D= dielectric displacement (charge perunit area)

d= piezoelectric coefficient

ε= dielectric constant

E= electric field strength

T= mechanical stress (force per unit area)

The indices l and m denote the direction components of dielectricdisplacement and field strength, respectively related to the orthogonalcrystal coordinates, from 1 to 3, corresponding to the axes X, Y, Z. Thecomponents of the elastic stress tensor are denoted analogously with thereduced indices μ. As usual T₁, T₂, T₃ denote pressures parallel to, andT₄, T₅, T₆ shear stresses about the axes X, Y, Z.

Thus, the piezolectric coefficient d_(l)μ constitutes a tensor of thethird order, which comprises 18 elements in the general case. If theauxiliary axes X, Y, Z, adopted as basic axes, coincide with elements ofsymmetry of the crystal, some tensor elements will be equal to 0 ormutually interdependent.

The present invention is concerned with crystals of the symmetry class32, i.e., with crystals of the trigonal-trapezohedral class for whichthe orthogonal auxiliary coordinate system is defined as having the Zaxis coinciding with the trigonal covering axis of rotation c and the Xaxis with the binary `a` axis. The Y axis if fixed as normal to the XZplane, so that a left-hand system of Cartesian coordinates results forlef-handed crystals and a right-hand one for right-hand crystals, cf.FIG. 1. With the crystal types preferred here, the contribution of thefield strength vector to the dielectric displacement is practicallynegligible compared with the influence of the elastic stress tensor.Hence, only the latter will be taken into account in the following.

Thus, the above relation (1) for the direct piezoelectric effect of acrystal of the symmetry described may be expressed in matrix form asfollows: ##EQU2## The piezoelectric d tensor thus contains only twoindependent elements, d₁₁ and d₁₄. Their magnitude andtemperature-dependence differ generally, and are governed by thecomposition and fine structure of the crystal in question. Consequently,a piezoelectric transducer element whose sensitivity is to responddifferently to temperature than the material constants d₁₁ and d₁₄ callsfor an orientation in which the sensitivity is a function of bothmaterial constants. To obtain a complete picture of all possibleorientations imaginable it is sufficient to state the formula for onetensor element to each of the four types of piezoelectric interaction.

If the position of the crystal element in relation to the coordinateaxis is described according to Gauss, by successive rotations about axesat right angles to each other, then two rotations are needed for the sixmatrix elements with l= μ (longitudinal effects) or l= (μ - 3) (synaxialshear effects), and three rotations for the other 12 elements. The firstrotation is performed about the X axis by the angle α (FIG. 2). Thesecond rotation by the angle β takes place about the Y' axis whichemerges from the first rotation (FIG. 3). The third rotation by theangle γ is performed about the doubly transformed axis Z" (FIG. 4). Theformulas that follow may be derived from the unreduced d matrix,(generally 27 elements with three-digit indices) by multiplying with theknown transformation matrices after taking the crystal symmetry intoaccount, and then reducing to two-digit indices.

For the longitudinal piezoelectric effect we have:

    d".sub.11 =d.sub.11 ·[cos.sup.2 β-3 sin.sup.2 α·sin.sup.2 β]·cos β. . . (3)

In this equation d₁₄ does not occur, which means that it is not possibleto produce a longitudinal piezoelectric transducer element from acrystal of symmetry 32 having a temperature response deviating from thatof the known X section. The synaxial shear sensitivity can likewise berepresented as an indicatrix with two rotations:

    d".sub.14 =2d.sub.11 cos α sin α(cos.sup.2 β-sin.sup.2 β)+ +d.sub.14 [(cos.sup.2 α-sin.sup.2 α) cos β-sin.sup.2 αsin.sup.2 β]. . .            (4)

The temperature coefficient depends on both angles of rotation. Theformulas for the transverse effect d'"₁₂ and for the antiaxial sheareffect d'"₂₆ with any orientation are:

    d'".sub.12 =(d.sub.14 cosαsinα-d.sub.11 cos.sup.2 α)·cosβcos.sup.3 γ +(4d.sub.11 cosαsinα+d.sub.14 cos.sup.2 α) cos βsin β cos.sup.2 ∛ sinγ+[d.sub.11 (2cos.sup.2 α+cos.sup.2 β-3sin.sup.2 αsin.sup.2 β)++d.sub.14 cos α sin α] cos β cos γ sin.sup.2 γ+(d.sub.14 cos.sup.2 α-2d.sub.11 cos α sinα) cosβsinβsinγ. . .                                                         (5)

    d'".sub.26 =12d.sub.11 cos α sin α cos β sin β cos.sup.2 γ sin γ+2 d.sub.11 [3(cos.sup.2 α-sin.sup.2 α)++(1+3sin.sup.2 α)cos.sup.2 β] cos β cos γ sin.sup.2 γ-(2d.sub.11 cos α+d.sub.14 sin αcos α cos β cos γ-(4d.sub.11 sin α+d.sub.14 cos αcos β sin β sin γ . . .                       (6)

With these formulas the magnitude of the piezoelectric coefficients andtheir temperature coefficient may be chosen freely, within the givenvariation range. A degree of freedom still remains allowing thesecondary sensitivities to be minimized within certain limits.

In general, crystal sections rotated threefold involve some difficultiesin practical use, such as manufacturing complications. More serious isthe fact that with such crystal elements of general orientation, all 18matrix elements differ from 0; this may lead to sensitivity to lateralforces and acceleration which is difficult to compensate. On account ofthis, the use of monoaxially rotated sections is recommended from theoutset where these are able to satisfy the requirements involved. Fromthe last two formulas it is possible to derive the expressions fororientations limited to one or two rotations. Where rotation takes placeonly about the X and Y' axes we obtain:

    d".sub.12 =(d.sub.14 sin α-d.sub.11 cos α) cos α cos β . . .                                              (7)

    d".sub.26 =-(2d.sub.11 cos α+d.sub.14 sin α) cos α cos β . . .                                              (8)

Obviously, in contrast to rotation about the X axis, rotation about theY' axis brings no alteration in the temperature-dependence of thepiezoelectric coefficients for the transverse and antiaxial effects, butonly an overall reduction in the sensitivity - which is normallyundesirable.

By analogy, where there is a first rotation about the X axis and asecond one about the transformed Z' axis we obtain the following:

    d".sub.12 =[d.sub.11 (1+2cos.sup.2 α)+d.sub.14 cos α sin α] cos γ-d.sub.11 (1+3cos.sup.2 α)cos.sup.3 γ. . . (9)

    d".sub.26 =-[2d.sub.11 (1+2cos.sup.2 α)+d.sub.14 cos α sin α] cos γ-2d.sub.11 (1+3cos.sup.2 α)·cos.sup.3 γ. . .                                              (10)

In this case α and γ are no longer independent, but it is easy to showthat here too, a rotation about the Z' axis leads to a loss ofsensitivity compared with a monoaxially rotated section with the samerelative temperature response. For the third special case, with thefirst rotation about Y and the second about Z', the following may bederived:

    d".sub.12 =d.sub.11 ·[(2+cos.sup.2 β)-(3+cos.sup.2 β)cos.sup.2 γ] cos β cos γ+d.sub.14 ·cos β sin β sin γ . . . tm (11)

    d".sub.26 =2d.sub.11 ·[(2+cos.sup.2 β)-(3+cos.sup.2 β)cos.sup.2 γ] cos β cos γ-d.sub.14 ·cos β sin β sin γ . . .                       (12)

Here the sine factors reduce the influence of d₁₄ if the angle ofrotation is small, while, if it is increased, the overall sensitivitydiminishes rapidly, so that this type of section is usually lessfavorable for measuring applications than that rotated about the X axis.

Thus, the foregoing considerations lead to the important conclusion thatof all the section orientations with a constant ratio of the d₁₁ and d₁₄increments for one of the tensor elements representing transverse orantiaxial shear effects, those orientations derived solely by rotationabout the X axis yield the greatest absolute value for the piezoelectriccoefficient. On account of the many potential applications thetransformed matrix of these purely X-rotated sections may be explainedmore fully: ##EQU3## in which

    d'.sub.11 =d.sub.11

    d'.sub.12 =-(d.sub.11 cos α-d.sub.14 sin α) cos α

    d'.sub.13 =-(d.sub.11 sin α+d.sub.14 cosα) sin α

    d'.sub.14 =+2d.sub.11 cos α·sin α30 d.sub.14 (cos.sup.2 α-sin.sup.2 α)

    d'.sub.25 =+(2d.sub.11 sin α-d.sub.14 cos α) cos α

    d'.sub.26 =-(2d.sub.11 cos α+d.sub.14 sin α) cos α

    d'.sub.35 =-(2d.sub.11 sin α-d.sub.14 cos α) sin α

    d'.sub.36 =+(2d.sub.11 cos α+d.sub.14 sin α) sin α

The three new elements included in the matrix can be related to theisotypic piezoelectric effects by substituting axes as follows:

    d 40 .sub.13 ( 60 )=+d'.sub.12 ( 60 ±90° )

    d'.sub.35 ( 60 )=+d 40 .sub.26 (α±90° )

    d'.sub.36 (α)=-d'.sub.25 (α±90° )

Two more important facts emerge from the matrix presentation. X-rotatedcrystal elements whose electrodes lie in the Y'Z' plane, i.e., inparticular the temperature-compensated transverse transducer elements,are insensitive to antiaxial shear forces since d'_(15=d') ₁₆ =0.X-rotated crystal elements whose electrodes lie parallel to the XZ' orXY' plane respond neither to longitudinal nor transverse pressure forcesnor to shear forces about the X axis. All these transducer elements arethus insensitive to disturbing lateral forces, just like the familiar Xand Y sections. This is decisive for their application in measuringcells of known design.

With the exception of d'₁₁, all components of the piezoelectric 'd"tensor differing from 0 represent linear combinations of d₁₁ and d₁₄terms. Accordingly, each of these piezoelectric effects can be madeindependent of temperature within a certain range by a suitable choiceof the orientation angle α. However, this aplies solely to crystalelements whose orientation is defined by a rotation about the X axis.With crystal sections rotated about the Y or Z axis, only piezoelectriceffects depending on a single coefficient exist. Thus, thetemperature-dependence of the sensitivity of these sections cannot bealtered with the orientation angle. In the case of exclusively Y-rotatedsections, 17 of the 18 matrix elements differ from 0; with Z-rotatedsections, 8 elements differ from 0. Consequently, only the X-rotatedcrystal sections need be further considered.

As an example, if the temperature coefficient of the first order ford'₁₂ is to be eliminated, the following conditional equation is obtainedfor the orientation angle: ##EQU4##

As a typical application, a quartz element for transverse pressureconversion may be described. With the data given below it is possible tocalculate any other quartz crystal transducer elements in similarfashion. The temperature behavior of the piezoelectric coefficient d₁₁has already been discussed at the outset with reference to FIG. 5. Alsoplotted in the same diagram is the basically different temperature curvefor d₁₄, whereas d₁₁, after a flat maximum at about -150° C., dropsprogressively with increasing temperature; d₁₄ has the opposite sign andits absolute value is progressively rising. At the high-low quartztransformation point of 573.3° C. where d₁₁ disappears, d₁₄ attains itsmaximum.

In order to replace these two curves by mathematical expressions withthe highest possible precision, five-term approximation polynomials ofthe following kind are very convenient:

    d.sub.11 =A.sub.o +A.sub.1 θ30 A.sub.2 θ.sup.2 30 A.sub.3 θ.sup.3 +A.sub.4 θ.sup.4 . . .                (17)

    d.sub.14 =B.sub.o +B.sub.1 θ+B.sub.2 θ.sup.2 +B.sub.3 θ.sup.3 +B.sub.4 θ.sup.4. . .                 (18)

a_(o) is the value of d₁₁ at which the temperature θ 0° C. ##EQU5## Thesame relations exist analogously between the B parameters and thedifferential quotients of d₁₄.

If Θ is substituted in degrees Celsius, in the temperature range -150to + 150° C., d₁₁ and d₁₄ are best reproduced by the following numericalvalues:

                  TABLE 1                                                         ______________________________________                                        A.sub.0 = + 2.280 · 10.sup.+1                                                     B.sub.0 = -6.433                                                                             pC · kp.sup.-1                           A.sub.1 = -4.436 · 10.sup.-3                                                      B.sub.1 = -1.242   10.sup.-2                                                                 pC · kp.sup.-1 · deg.sup.-1                                 2                                                 A.sub.2 = -9.790 · 10.sup.-6                                                      B.sub.2 = +7.232 · 10.sup.-6                                                        pC · kp.sup.-1 · deg.sup.-2     A.sub.3 = +5.551 · 10.sup.-8                                                      B.sub.3 = -1.609 · 10.sup.-8                                                        pC · kp.sup.-1 · deg.sup.-3     A.sub.4 = -1.263 · 10.sup.-10                                                     B.sub.4 = -4.053 · 10.sup.-11                                                       pC · kp.sup.-1 · deg.sup.-5     ______________________________________                                    

By substituting these polynomials into the transformation equations ofthe matrix elements, all eight piezoelectric coefficients of a quartzelement can be calculated as a function of the temperature for anyorientation angle α. The results of this calculation are plottedgraphically in FIG. 6 for the transverse effect--d' ₁₂. As the angle αincreases, the temperature curves initially become flatter in the rangeof about 0° to about 300° C., after which they assume a more convexshape. The maximum sensitivity then shifts to higher temperatures, whileat the same time the height of the peak declines monotonically. In FIG.7 the temperature of this sensitivity maximum for the transverseeffect--d'₁₂ is plotted in the form of a solid curve as a function ofthe angle of rotation α.

Ideally suited for measuring applications is that quartz section whichhas a constant and, at the same time, highest possible sensitivitywithin the accuracy requirements over the widest possible temperatureinterval. The orientation of this desired section may be ascertained byvariation calculus or approximation. If the requirement for constancy ofthe calibration factor is fixed at ± 1%, then the upper limit of thepiezoelectric sensitivity will be its maximum, while the lower limitwill correspond to two real roots, which generally have differentintervals from the maximum. These two temperatures are also plotted inFIG. 7, in the form of broken lines.

The temperature interval within which the piezoelectric sensitivityremains constant within ± 1% is equal to the distance between these twocurves for every value of the angle of rotation. The magnitude of thistemperature interval is shown in FIG. 8 as a function of α. From this itfollows that the quartz section with the best temperature-independencehas an orientation angle of about 23.9°. Its transverse sensitivity is-21.337 pC/kg to within ± 1% between -143 and +408° C., so that theinterval amounts to about 550°. If the precision requirement is narrowedto ± 0.5%, the temperature range becomes approximately -93 to +369° C.,in other words, substantially 462 degrees .

In the technologically most important temperature range, with thissection, the temperature response of the piezoelectric coefficient--d'₁₂is practically no longer measurable, since it deviates by only ± 1% fromthe value of -21.423 pC/kg between approximately -10 and ±294° C. Thecomplete temperature curve for this transverse section is plotted inFIG. 9, together with the associated longitudinal sensitivity d₁₁, whosecurve corresponds to that of the usual quartz sections. It will be seenfrom the diagram that the piezoelectric coefficient--d' ₁₂ of thesection in accordance with the invention is only slightly smaller thanthe familiar X section below approximately 240° C. Above 240° C. thesensitivity of the proposed section is even better than that of the Xsection. Another favorable point is that at the high-low transformationpoint of quartz (573.3° C.) the piezoelectric effect of the new sectiondoes not vanish as with the known sections, but still amounts to 6.88pC/kp.

In addition to the improvements described above, the choice of sectionorientation according to the invention brings a further decisive gain.If a quartz element with an orientation angle close to the value statedis preloaded in the direction of its transformed Y' axis, this causes areduction of the free enthalpy of the crystal lattice and a commensuraterise in the energy level for the twinning lattice state in accordancewith the so-called Dauphine law. This can be proved thermodynamically bythe tensor-based directional dependence of the elasticity coefficientsof quartz. Thus an energetic instability for twin domains results and,consequently, a greatly reduced proneness to twinning. This makes itpossible to employ quartz elements conforming to the invention attemperatures where all conventional quartz transducers would becomeunserviceable in a short time.

The quartz section described herein represents only an arbitrarilyselected example of the application of the invention. With the datagiven above, other section orientations may be ascertained as required,for transducer quartzes to be employed at particularly high temperaturesfor instance, or for transducers with 0 temperature coefficient at roomtemperature instead of around 140° C. etc. Also, quartz elements forconverting antiaxial shear forces may be calculated analogously, bystarting from the equations for d'₂₆, instead of d'₁₂. The temperaturecurve of this piezoelectric coefficent d'₂₆ is plotted in FIG. 10 fordifferent values of the orientation angle α. It will be seen that, inthis case, a reduction of the temperature coefficients calls for arotation in the opposite direction to that for the transversepiezoelectric coefficient d'₁₂, and that the associated reduction in themean sensitivity is much more pronounced. For most practical purposes itwill, therefore, be best to compromise between sensitivity andindependence of temperature. Depending on the operating temperature,orientations between about 120° and about 160° will generally beparticularly suitable. Of course, these sections may be calculated bythe other methods described as well; further details on this aresuperfluous and can be readily calculated by one skilled in the art fromthe above discussion.

It should also be pointed out that the piezoelectric sensitivity of acomplete transducer generally does not yield exactly the sametemperature-dependence as that of the crystal element per se. Thiscircumstance results primarily from the division of the forces acting onthe measuring cell between the crystal elements on the one hand andpreloading device on the other. When the temperature changes, these twoparts expand by different amounts depending on their dimensions andcoefficients of expansion. In this way, the load distribution betweencrystal elements and a preloading device is altered. A difference in thetemperature-dependence of the respective coefficients of elasticity alsocontributes to this Moreover, other effects like thetemperature-dependence of the dielectric constants etc. may exercise aninfluence. For accurate predictions, however, at least the piezoelectricand elastic constants with their temperature coefficients, thecoefficients of thermal expansion and the geometrical dimensions of thecomponents must be taken into account.

From this it follows that the temperature-dependence of thepiezoelectric sensitivity of a transducer can be modified within certainlimits by the dimensioning and choice of material constants, that is,irrespective of the crystal orientation. But the effects generallymentioned have relatively little influence, compared with the crystalorientation. Depending on the dimensions it may even disappearaltogether, as has been assumed in the typical application illustrated.

Applications for piezoelectric transducers may also be imagined callingfor designs with two differently oriented types of crystal elements.These include, for example, acceleration-compensated pressure and forcemeasuring cells, in which the compensating elements must have only afraction, such as a third of the sensitivity of the main elements butotherwise the same relative temperature-dependence. In such cases theuse of two-fold rotated crystal elements is generally preferably tocrystal forms with partly short-circuited electrode areas. According torequirements of lateral sensitivity and other effects, the orientationmay be chosen in a different way. In the case of transverse transducers,for example, it may be convenient to perform a first rotation about theY or Z axis as the case may be, followed by a second rotation about thetransformed X' axis.

Finally it should be emphasized that the invention is not confined toquartz crystals but may be applied analogously to other materials of thecrystal class 32 as well. At the present time, however, not manycrystals of this symmetry are known which can also be grown easily inlarge species and which fulfill the exacting requirements in the way ofmechanical, piezoelectric and electrical properties. For applications atparticularly high temperatures and other extreme conditions, thetrigonal phases of germanium dioxide GeO₂ and aluminum phosphate AlPO₄,for example, offer certain possibilities.

I claim:
 1. A piezoelectric crystal element for use in force andpressure transducers and accelerometers converting forces, pressures,torques and accelerations into electrical signals such that thetemperature dependence of at least one of the transverse and antiaxialshear sensitivity coefficients is considerably reduced over apredetermined temperature interval, as compared to the longitudinalpiezoelectric coefficient,said element comprising a member ofmonocrystalline material of symmetry class 32, having two forceintroduction surfaces essentially parallel to a crystallographic`a`-axis but inclined with respect to the `c`-axis so as to intersectthe latter at an orientation angle ##EQU6## wherein d₁₁ and d₁₄ are theindependent coefficients of the piezoelectric d-tensor, and `θ`represents the temperature at which said crystal element is required toyield a maximum value of piezoelectric sensitivity and a zerotemperature coefficient of sensitivity, and wherein `f` is a constanthaving a value of +1 for the transverse piezolectric effect and a valueof -2 for the antiaxial shear piezoeffect.
 2. A piezoelectric crystalelement according to claim 1, wherein said monocrystalline material isquartz, and wherein the transverse piezoelectric sensitivity has amaximum value with a zero temperature coefficient at a temperaturespecified within the range from about -63° C. to about +450° C., fromwhich results an orientation angle α having a value between about 10 andabout 40°.
 3. A piezoelectric crystal element according to claim 1,wherein said monocrystalline material is quartz and wherein theantiaxial shear sensitivity has a maximum value with a substantiallyzero temperature coefficient at a temperature within the range fromabout -63° C. to about +450° C., from which results an orientation angleα having a value of between about 120 and about 160 degrees.
 4. Apiezoelectric crystal element for use in force and pressure transducersand accelerometers converting forces, pressures, torques andaccelerations into electrical signals, comprising a member ofmonocrystalline material, the symmetry of which corresponds to that ofthe piezolectric d coefficient tensor of the point group 32, and whereintwo parallel force introduction surfaces are essentially parallel to acrystallographic `a` axis and are inclined with respect to the `c` axis,so that the temperature dependence of the piezoelectric sensitivity forone of (i) transverse and (ii) shear effect is considerably reduced overa predetermined temperature interval, as compared to the longitudinalpiezoelectric coefficient,characterized in that the surfaces of thecrystal at which forces are introduced have an orientation angle α(i)either of from about 10° to about 40° with respect to the `c`-axis, andits complement, whereby the temperature dependence of at least one ofthe piezoelectric coefficients d'₁₂ and d'₁₃, respectively, is minimizedover a temperature interval in the range from about -143° C. to about+408° C., or (ii) from about 120° to about 160° with respect to the`c`-axis, and its complement, whereby the temperature dependence of atleast one of the piezoelectric coefficients d'₂₆ and d'₃₅, respectively,is minimized over a temperature interval in the range from about -200°C. to about +400° C.
 5. A piezoelectric crystal element for use in forceand pressure transducers and accelerometers coverting forces, pressures,torqued and accelerations into electrical signals such that thetemperature dependence of at least one of the transverse and antiaxialshear sensitivity coefficients is considerably reduced over apredetermined temperature interval, as compared to the longitudinalpiezoelectric coefficient,said element comprising a member ofmonocrystalline material of symmetry class 32, having two forceintroduction surfaces essentially parallel to a crystallographic`a`-axis but inclined with respect to the `c`-axis so as to intersectthe latter at an orientation angle ##EQU7## wherein d₁₁ and d₁₄ are theindependent coefficients of the piezoelectric d-tensor, and `θ`represents the temperature at which said crystal element is required toyield a maximum value of piezoelectric sensitivity and a substantiallyzero temperature coefficient of sensitivity, and wherein `f` is aconstant having a value of +1 for the transverse piezoelectric effectand a value of -2 for the antiaxial shear piezoeffect.
 6. Apiezoelectric crystal element according to claim 5, wherein said angleα, at which said two force introduction surfaces intersect saidcrystallographic `c` axis if from about 10° to about 40° such that thetemperature dependence of the piezoelectric d-tensor coefficient d'₁₂for the transverse piezoelectric sensitivity is minimized over saidpredetermined temperature interval ranging from about -143° C. to about+408° C.
 7. A piezoelectric crystal element according to claim 6,wherein said angle α is 23.9° such that the transverse piezoelectricsensitivity is constant to within about ± 1% over said temperatureinterval of about -143° C. to about +408° C.
 8. A piezoelectric crystalelement according to claim 7, wherein said transverse piezoelectricsensitivity is constant to within ± 0.5% over a temperature interval offrom about -93° C. to about ±369° C.
 9. A piezoelectric crystal elementaccording to claim 8, wherein said transverse piezoelectric sensitivityis constant to within ± 0.1% over a temperature interval of from about-10° C. to about +294° C.
 10. A piezoelectric crystal element accordingto claim 5, comprising a member of monocrystalline quartz wherein twoforce introduction surfaces are essentially parallel to acrystallograhic `a`-axis and are inclined with respect to the `c`-axisso that the temperature dependence of the piezoelectric sensitivity forone of (i) transverse and (ii) shear effect is considerably reduced overa predetermined temperature interval, as compared to the longitudinalpiezolectric coefficient, characterized in that the surfaces of thecrystal at which forces are introduced have an orientation angle αeither (i) of from about 10° to about 40° with respect to the `c`-axis,and its complement, whereby the temperature dependence of at least oneof the piezoelectric coefficients d'₁₂ and d'₁₃, respectively, isminimized over a temperature interval in the range from about -143° C.to about 30 408° C., or (ii) of from about 110° to about 155° withrespect to the `c`-axis, and its complement, whereby the temperaturedependence of at least one of the piezoelectric coefficients d'₂₆ andd'₃₅, respectively, is minimized over a temperature interval in therange of from about -200° C. to about +400 ° C.